Optimal. Leaf size=69 \[ \frac{\sqrt{b x^2+c x^4} (2 b B-A c)}{b c^2 x}-\frac{x^3 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.149797, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2037, 1588} \[ \frac{\sqrt{b x^2+c x^4} (2 b B-A c)}{b c^2 x}-\frac{x^3 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2037
Rule 1588
Rubi steps
\begin{align*} \int \frac{x^4 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{(b B-A c) x^3}{b c \sqrt{b x^2+c x^4}}+\frac{(2 b B-A c) \int \frac{x^2}{\sqrt{b x^2+c x^4}} \, dx}{b c}\\ &=-\frac{(b B-A c) x^3}{b c \sqrt{b x^2+c x^4}}+\frac{(2 b B-A c) \sqrt{b x^2+c x^4}}{b c^2 x}\\ \end{align*}
Mathematica [A] time = 0.0229948, size = 35, normalized size = 0.51 \[ \frac{x \left (-A c+2 b B+B c x^2\right )}{c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -Bc{x}^{2}+Ac-2\,Bb \right ){x}^{3}}{{c}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21333, size = 53, normalized size = 0.77 \begin{align*} \frac{{\left (c x^{2} + 2 \, b\right )} B}{\sqrt{c x^{2} + b} c^{2}} - \frac{A}{\sqrt{c x^{2} + b} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27892, size = 88, normalized size = 1.28 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left (B c x^{2} + 2 \, B b - A c\right )}}{c^{3} x^{3} + b c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33152, size = 81, normalized size = 1.17 \begin{align*} -\frac{2 \, B \sqrt{b}}{{\left ({\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c\right )} c} + \frac{B b - A c}{\sqrt{c + \frac{b}{x^{2}}} c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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